As indicated in Section REF , small cut elements do not only adversely affect the conditioning of the resulting linear system, but can also disturb the quality of the solution as the normal gradient is not adequately controlled. When Nitsche's method is applied on a boundary-fitted discretization, the trace inverse inequality
\(\Vert  \partial _n u_h \Vert _{T \cap \partial \Omega _D} \lesssim h_T^{-1/2} \Vert  \nabla u_h \Vert _{T}\) 
holds on every element \(T\)  adjacent to the Dirichlet boundary \(\partial \Omega _D\) ; see, e.g., [1]}. By virtue of this trace inequality, the bilinear operator \(a_h(\cdot ,\cdot )\)  defined in (REF ) is coercive with a global Nitsche parameter \(\beta \)  that scales with \(h^{-1}\) , and optimal error bounds with respect to the \(H^1(\Omega )\) -norm hold. With an immersed discretization, however, an element \(T\)  that intersects \(\partial \Omega _D\)  only partially intersects the physical domain \(\Omega \) . Therefore, the aforementioned trace inverse inequality assumes the form
\(\Vert  \partial _n u_h \Vert _{T \cap \partial \Omega _D} \lesssim h_{T_\Omega }^{-1/2} \Vert  \nabla u_h \Vert _{T_\Omega }\) ,
where \(h_{T_\Omega }\)  indicates a generalized thickness of the fragment \(T_\Omega = T \cap \Omega \)  (the intersection of element \(T\)  and the physical domain) normal to \(T \cap \partial \Omega _D\)  (the intersection of element \(T\)  and the Dirichlet boundary); see Section REF . Consequently, coercivity of the bilinear form \(a_h(\cdot ,\cdot )\)  requires an element-wise Nitsche parameter \(\beta \)  that is
not inversely proportional to the characteristic mesh width of the background element, \(h_T\) , but instead is inversely proportional to the generalized thickness of the intersection, i.e., \(\beta \sim h_{T_\Omega }^{-1} \gtrsim h_T^{-1}\) . Due to the fact that without special precautions, in immersed finite element methods one has no control over \(h_{T_\Omega }\) , the Nitsche parameter can in principle become arbitrarily large.
